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Irénée Briquel, Pascal Koiran
To cite this version:
Irénée Briquel, Pascal Koiran. A Dichotomy Theorem for Polynomial Evaluation. Mathematical
Foun-dations of Computer Science 2009, Aug 2009, Novy Smokovec, Slovakia. pp.187-198,
�10.1007/978-3-642-03816-7�. �ensl-00360974v2�
Evaluation
Ir´en´ee Briquel and Pascal Koiran
LIP⋆
, ´Ecole Normale Sup´erieure de Lyon, Universit´e de Lyon {irenee.briquel,pascal.koiran}@ens-lyon.fr
Abstract. A dichotomy theorem for counting problems due to Creignou and Hermann states that or any finite set S of logical relations, the count-ing problem #SAT(S) is either in FP, or #P-complete. In the present paper we show a dichotomy theorem for polynomial evaluation. That is, we show that for a given set S, either there exists a VNP-complete family of polynomials associated to S, or the associated families of polynomi-als are all in VP. We give a concise characterization of the sets S that give rise to “easy” and “hard” polynomials. We also prove that several problems which were known to be #P-complete under Turing reductions only are in fact #P-complete under many-one reductions.
Key words: Algebraic complexity, polynomial evaluation, Valiant’s model, Constraint Satisfaction Problems, dichotomy theorem, #P-completeness
1
Introduction
.
In a seminal paper, Schaefer [12] proved a dichotomy theorem for boolean constraint satisfaction problems: he showed that for any finite set S of logi-cal relations the satisfiability problem SAT(S) for S-formulas is either in P, or NP-complete. Here, an S-formula over a set of n variables is a conjunction of relations of S where the arguments of each relation are freely chosen among the n variables. Schaefer’s result was subsequently extended in a number of direc-tions. In particular, dichotomy theorems were obtained for counting problems, optimization problems and the decision problem of quantified boolean formulas. An account of this line of work can be found in the book by Creignou, Khanna and Sudan [5]. In a different direction, constraint satisfaction problems were also studied over non-boolean domains. This turned out to be a surprisingly difficult question, and it took a long time before a dichotomy theorem over domains of size 3 could be obtained [1].
In the present paper we study polynomial evaluation from this dichotomic point of view. We work within Valiant’s algebraic framework: the role of the com-plexity class NP in Schaefer’s dichotomy theorem will be played by the class VNP of “easily definable” polynomial families, and the role of P will be played by the
⋆
class VP of “easily computable” polynomial families [13,3]. There is a well-known connection between counting problems and polynomial evaluation. For instance, as shown by Valiant the permanent is complete in both settings [14,13]. In the realm of counting problems, a dichotomy theorem was obtained by Creignou and Hermann [4,5].
Theorem 1. For any finite set S of logical relations, the counting problem #SAT(S) is either in FP, or #P-complete.
In fact, the sets S such that #SAT(S) is in FP are exactly the sets containing only affine constraints (a constraint is called affine if it expressible as a system of linear equations over Z/2Z).
Main Contributions
To a family of boolean formulas (φn) we associate the multilinear polynomial
family P (φn)(X) = X ε φn(ε)X ε , (1)
where Xε is the monomial Xε1
1 · · · X εk(n)
k(n) , and k(n) is the number of variables
of φn. Imagine that the φn are chosen among the S-formulas of a fixed finite set
S of logical relations. One would like to understand how the complexity of the polynomials P (φn) depends on S.
Definition 1. A family (φn) of S-formulas is called a p-family if φn is a
con-junction of at most p(n) relations from S, for some polynomial p (in particular,
φn depends on polynomially many variables when S is finite).
Theorem 2 (Main Theorem). Let S be a finite set of logical relations. If S
contains only affine relations of at most two variables, then the families (P (φn))
of polynomials associated to p-families of S-formulas (φn) are in VP. Otherwise,
there exists a p-family (φn) of S-formulas such that the corresponding polynomial
family P (φn) is VNP-complete.
We can remark, that the hard cases for counting problems are strictly in-cluded in our hard evaluation problems, exactly as the hard decision problems in Schaefer’s theorem were strictly included in the hard counting problems.
In our algebraic framework the evaluation of the polynomial associated to a given formula consists in solving a “weighted counting” problem: each assignment (ε1, . . . , εk) of the variables of φ comes with a weight X1ε1· · · Xkεk. In particular,
when the variables Xiare all set to 1, we obtain the counting problem #SAT(S).
It is therefore natural that evaluation problems turn out to be harder than counting problems.
The remainder of this paper is mostly devoted to the proof of Theorem 2. Allong the way, we obtain several results of independent interest. First, we obtain several new VNP-completeness results. The main ones are about:
(i) the vertex cover polynomial VCP(G) and the independent set polynomial IP(G), associated to a vertex-weighted graph G. Most VNP-completeness results in the literature (and certainly all the results in Chapter 3 of [3]) are about edge-weighted graphs.
(ii) the antichain polynomial AP(X) and the ideal polynomial IPP(X), associ-ated to a weighted poset (X, ≤).
Unlike in most VNP-completeness results, we need more general reductions to establish VNP-completeness results than Valiant’s p-projection. In Section 6, we use the “c-reductions”, introduced by B¨urgisser [2,3] in his work on VNP families that are neither p-computable nor VNP-complete. They are akin to the oracle (or Turing) reductions from discrete complexity theory. The c-reduction has not been used widely in VNP-completeness proofs. The only examples that we are aware of are:
(i) A remark in [3] on probability generating functions.
(ii) The VNP-completeness of the weighted Tutte polynomial in [10]. Even there, the power of c-reductions is used in a very restricted way since a single oracle call is performed in each reduction.
By contrast, the power of oracle reductions has been put to good use in #P-completeness theory (mostly as a tool for performing interpolation). Indeed, as pointed out in [8], “interpolation features prominently in a majority of completeness proofs”, and “it is not clear whether the phenomenon of #P-completeness would be as ubiquitous if many-one reducibility were to be used in place of Turing.” We argue that the importance of Turing reductions in #P-completeness should be revised downwards since, as a byproduct of our VNP-completeness results, we can replace Turing reductions by many-one reductions in several #P-completeness results from the literature. In particular, we obtain a many-one version of Creignou and Hermann’s dichotomy theorem1. We leave
it as an open problem whether the 0/1 partial permanent is #P-complete under many-one reductions (see Section 3 for a definition of the partial permanent, and [7] for a # P-completeness proof under oracle reductions).
Organization of the Paper and Additional Results
Earlier in this section we gave an informal introduction to constraint satisfac-tion problems. We give more precise definisatisfac-tions at the beginning of Secsatisfac-tion 2. The remainder of that section is devoted to Valiant’s algebraic model of com-putation. We also deal briefly with the easy cases of Theorem 2 (Remark 1). We then establish the proof of the hard cases of Theorem 2, beginning with
1 Many-one reductions (Definition 2) are called many-one counting reductions in [4,5].
It was already claimed in [4,5] that Theorem 1 holds true for many-one reductions. This was not fully justified since the proof of Theorem 1 is based on many-one reductions from problems which were previously known to be #P-complete under oracle reductions only. The present paper shows that this claim was indeed correct.
the case of non affine constraints. For that case, the high-level structure of the proof is similar to Creignou and Hermann’s proof of #P-completeness of the corresponding counting problems in [4]. The singletons S = {OR2}, S = {OR1}
and S = {OR0} play a special role in the proof. Here OR2denotes the negative
two-clause (x, y) 7→ (x ∨y); OR0denotes the positive two-clause (x, y) 7→ (x∨y);
and OR1 denotes the implicative two-clause (x, y) → (x ∨ y). The
correspond-ing VNP-completeness results for S = {OR2} and S = {OR0} are established
in section 3; the case of {OR1} is treated in Section 4.These results are put
together with Creignou and Hermann’s results in Section 5 to establish the ex-istence of a VNP-complete family for any set S containing non affine clauses (Theorem 8). Section 6 deals with the affine clauses with at least three variables (Theorem 9). This completes the proof of Theorem 2. In Section 7, we build on our VNP-completeness results to prove #P-completeness under many-one re-ductions for several problems which were only known to be #P-complete under oracle reductions.
2
Preliminaries
2.1 Constraint satisfaction problems
We define a logical relation to be a function from {0, 1}k to {0, 1}, for some
integer k called the rank of the relation. Let us fix a finite set S = {φ1, . . . , φn}
of logical relations. An S-formula over n variables (x1, . . . , xn) is a conjunction of
boolean formulas, each of the form gi(xji(1), . . . , xji(ki)) where each gibelongs to
S and ki is the rank of gi. In words, each element in the conjunction is obtained
by applying a function from S to some variables chosen among the n variables. An instance of the problem SAT(S) studied by Schaefer [12] is an S-formula φ, and one must decide whether φ is satisfiable. For instance, consider the 3 boolean relations OR0(x, y) = x∨y, OR1(x, y) = x∨y and OR2(x, y) = x∨y. The
classical problem 2-SAT is SAT(S) where S = {OR0, OR1, OR2}. The counting
problem #SAT(S) was studied by Creignou and Hermann [4]. In this paper we study the complexity of evaluating the polynomials P (φ) in (1). We establish which sets S give rise to VNP-complete polynomial families, and which one give rise only to easy to compute families. We next define these notions precisely.
2.2 #P-completeness and VNP-completeness
Let us introduce the notion of many-one reduction for counting problems [16]: Definition 2 (Many-one reduction). Let f : {0, 1}∗→ N and g : {0, 1}∗→
Nbe two counting problems. A many-one reduction from f to g consists of a pair
of polynomial-time computable functions σ : {0, 1}∗ → {0, 1}∗ and τ : N → N
such that for every x ∈ {0, 1}∗, the equality f (x) = τ (g(σ(x))) holds. When τ is
A counting problem f is #P-hard under many-one reduction if every problem in #P admits a many-one reduction to f .
In Valiant’s model one studies the computation of multivariate polynomials. This can be done over any field. In the sequel we fix a field K of characteristic 6= 2. All considered polynomials are over K.
A p-family is a sequence f = (fn) of multivariate polynomials such that
the number of variables and the degree are polynomially bounded functions of n. A prominent example of a p-family is the permanent family PER = (PERn),
where PERnis the permanent of an n×n matrix with independent indeterminate
entries.
We define the complexity of a polynomial f to be the minimum number L(f ) of nodes of an arithmetic circuit computing f . We recall that the internal nodes of an arithmetic circuit perform additions or multiplications, and each input node is labeled by a constant from K or a variable Xi.
Definition 3 (VP). A p-family (fn) is p-computable if L(fn) is a polynomially
bounded function of n. Those families constitute the complexity class VP.
In Valiant’s model, VNP is the analogue of the class NP (or perhaps more accurately, of #P).
Definition 4 (VNP). A p-family (fn) is called p-definable if there exists a
p-computable family g = (gn) such that
fn(X1, . . . , Xp(n)) =
X
ε∈{0,1}q(n)
gn(X1, . . . , Xp(n), ε1, . . . , εq(n))
The set of p-definable families forms the class VNP.
Clearly, VP is included in VNP. To define VNP-completeness we need a notion of reduction:
Definition 5 (p-projection). A polynomial f with v arguments is said to be
a projection of a polynomial g with u arguments, and we denote it f ≤ g, if
f (X1, . . . , Xv) = g(a1, . . . , au) where each ai is a variable of f or a constant
from K.
A p-family (fn) is a p-projection of (gm) if there exists a polynomially bounded
function t : N → N such that: ∃n0∀n ≥ n0, fn≤ gt(n).
Definition 6 (VNP-completeness). A p-family g ∈ VNP is VNP-complete
if every p-family f ∈ VNP is a p-projection of g.
The VNP-completeness of the permanent under p-projections [13,3] is a cen-tral result in Valiant’s theory.
It seems that p-projections are too weak for some of our completeness results. Instead, we use the more general notion of c-reduction [2,3]. First we recall the notion of oracle computation :
Definition 7. The oracle complexity Lg(f ) of a polynomial f with respect to the
oracle polynomial g is the minimum number of arithmetic operations (+, ∗) and evaluations of g over previously computed values that are sufficient to compute
f from the indeterminates Xi and constants from K.
Definition 8 (c-reduction). Let us consider two p-families f = (fn) and g =
(gn). We have a polynomial oracle reduction, or c-reduction, from f to g (denoted
f ≤c g) if there exists a polynomially bounded function t : N → N such that the
map n 7→ Lgt(n)(f
n) is polynomially bounded.
We can define a more general notion of VNP-completeness based on c-reductions: A p-family f is VNP-hard if g ≤c f for every p-family g ∈ VNP.
It is VNP-complete if in addition, f ∈ VNP. The new class of VNP-complete families contains all the classical VNP-complete families since every p-reduction is a c-reduction.
In our completeness proofs we need c-reductions to compute the homogeneous components of a polynomial. This can be achieved thanks to a well-known lemma (see e.g. [3]):
Lemma 1. Let f be a polynomial in the variables X1, . . . , Xn. For any δ such
that δ ≤ deg f , let denote f(δ) the homogeneous component of degree δ of f .
Then, Lf(f(δ)) is polynomially bounded in the degree of f .
By Valiant’s criterion (Proposition 2.20 in [3]), for any finite set S of logical relations and any p-family (φn) of S-formulas the polynomials (P (φn)) form a
VNP family. Furthermore, the only four boolean affine relations with at most two variables are (x = 0), (x = 1), (x = y) and (x 6= y). Since for a conjunction of such relations, the variables are either independent or completely bounded, a polynomial associated to a p-family of such formulas is immediately factorizable. Thus :
Remark 1. For a set S of affine relations with at most two variables, every
p-family of polynomials associated to S-formulas is in VP.
All the work in the proof of Theorem 2 therefore goes into the hardness proof.
3
Monotone 2-clauses
In this section we consider the set {OR2} = {(x, y) 7→ (x ∨ y)} and {OR0} =
{(x, y) 7→ (x ∨ y)}. For S = {OR2} and S = {OR0}, we show that there exists
a VNP-complete family of polynomials (P (φn)) associated to a p-family of
S-formulas (φn).
The partial permanent PER∗(A) of a matrix A = (Ai,j) is defined by the
formula: PER∗(A) =X π Y i∈defπ Aiπ(i)
where the sum runs over all injective partial maps from [1, n] to [1, n]. It is shown in [3] that the partial permanent is VNP-complete (the proof is attributed to
Jerrum). The partial permanent may be written as in (1), where φnis the boolean
formula that recognizes the matrices of partial maps from [1, n] to [1, n]. But φn
is a p-family of {OR2}-formulas since
φn(ε) = ^ i,j,k:j6=k εij∨ εik∧ ^ i,j,k:i6=k εij∨ εkj.
Here the first conjunction ensures that the matrix ε has no more than one 1 on each row; the second one ensures that ε has no more than one 1 on each column. We have obtained the following result.
Theorem 3. The family (φn) is a p-family of {OR2}-formulas, and the
poly-nomial family (P (φn)) is VNP-complete under p-projections.
The remainder of this section is devoted to the set S = {OR0} = {(x, y) 7→
x ∨ y}. The role played by the partial permanent in the previous section will be played by vertex cover polynomials. There is more work to do because the corresponding VNP-completeness result is not available from the literature.
Consider a vertex-weighted graph G = (V, E): to each vertex vi ∈ V is
associated a weight Xi. The vertex cover polynomial of G is
VCP(G) =X
S
Y
vi∈S
Xi (2)
where the sum runs over all vertex covers of G (recall that a vertex cover of G is a set S ⊆ V such that for each edge e ∈ E, at least one of the two endpoints of e belongs to S). The univariate vertex cover polynomial defined in [6] is a specialization of ours; it is obtained from VCP(G) by applying the substitutions Xi:= X (for i = 1, . . . , n), where X is a new indeterminate.
Our main result regarding {OR0}-formulas is as follows.
Theorem 4. There exists a family Gn of polynomial size bipartite graphs such
that:
1. The family (VCP(Gn)) is VNP-complete.
2. VCP(Gn) = P (φn) where φn is a p-family of {OR0}-formulas.
Given a vertex-weighted graph G, let us associate to each vi ∈ V a boolean
variable εi. The interpretation is that vi is chosen in a vertex cover when εi is
set to 1. We then have
VCP(G) = X ε∈{0,1}|V | ^ (vi,vj)∈E εi∨ εjX ε .
The second property in Theorem 4 will therefore hold true for any family (Gn)
of polynomial size graphs.
To obtain the first property, we first establish a VNP-completeness result for the independent set polynomial IP(G). This polynomial is defined like the vertex cover polynomial, except that the sum in (2) now runs over all independent sets S (recall that an independent set is a set S ⊆ V such that there are no edges between any two elements of S).
Theorem 5. There exists a family (G′
n) of polynomial size graphs such that
IP(G′
n) = PER∗n where PER∗n is the n × n partial permanent. The family IP(G′n)
is therefore VNP-complete. Proof. The vertices of G′
n are the n2 edges ij of the complete bipartite graph
Kn,n, and the associated weight is the indeterminate Xij. Two vertices of G′n
are connected by an edge if they share an endpoint in Kn,n. An independent set
in G′
n is nothing but a partial matching in Kn,n, and the corresponding weights
are the same.
Next we obtain a reduction from the independent set polynomial to the vertex cover polynomial. The connection between these two problems is not astonishing since vertex covers are exactly the complements of independent sets. But we deal here with weighted counting problems, so that there is a little more work to do. The connection between independent sets and vertex covers does imply a relation between the polynomials IP(G) and VCP(G). Namely,
IP(G)(X1, . . . , Xn) = X1· · · Xn· VCP(G)(1/X1, . . . , 1/Xn). (3) Indeed, IP(G) = X S independent X1· · · Xn Q vi6∈SXi = X1· · · Xn X S′ vertex cover 1 Q vi∈S′Xi . Recall that the incidence graph of a graph G′= (V′, E′) is a bipartite graph
G = (V, E) where V = V′∪ E′. In the incidence graph there is an edge between
e′ ∈ E′ and u′ ∈ V′ if u′ is one of the two endpoints of e′ in G. When G′ is
vertex weighted, we assign to each V′-vertex of G the same weight as in G and
we assign to each E′-vertex of G the constant weight −1.
Lemma 2. Let G′be a vertex weighted graph and G its vertex weighted incidence
graph as defined above. We have the following equalities:
VCP(G) = (−1)e(G′)IP(G′) (4)
IP(G) = (−1)e(G′)VCP(G′) (5)
where e(G′) is the number of edges of G′.
Proof. We begin with (4). To each independent set I′ of G′ we can injectively
associate the vertex cover C = I′∪ E′. The weight of C is equal to (−1)e(G′)
times the weight of I′. Moreover, the weights of all other vertex covers of G
add up to 0. Indeed, any vertex cover C which is not of this form must contain two vertices u′, v′∈ V′ such that u′v′ ∈ E′. The symmetric difference C∆{u′v′}
remains a vertex cover of G, and its weight is opposite to the weight of C since it differs from C only by a vertex u′v′ of weight −1.
The equality (5) follow from the combination of (3) and (4).
To complete the proof of Theorem 4 we apply Lemma 2 to the graph G′= G′ n of
Theorem 5. The resulting graph G = Gn satisfies VCP(Gn) = IP(G′n) = PER∗n
since G′
4
Implicative 2-clauses
Here we consider the set S = {OR1} = {(x, y) → (x∨y)}. Those logical relations
are called implicative, because x∨y is equivalent to y ⇒ x. The #P-completeness of #SAT(S) was established by a chain of reductions in [11] and [9]. Here we will follow this chain of reductions to find a VNP-complete family associated to S-formulas. These two articles show consecutively that the problems of counting the independent sets, the independent sets in a bipartite graph, the antichains in partial ordered sets (posets), the ideals in posets, and finally satisfaction of implicative 2-clauses are #P-complete. We will start from the family (G′
n) such
that IP(G′
n) = PER∗n, whose existence is stated in Theorem 5, and follow the
reductions for the counting problems. We first transform the family (G′
n) into a family of bipartite graphs, without
changing the independent set polynomials.
Lemma 3. There exists a family of bipartite graphs (G′′
n) such that IP(G′′n) =
PER∗n, the partial permanent of size n × n.
Proof. By Lemma 2, we know how to transform a graph G into a bipartite graph
G′ such that VCP(G′) = (−1)e(G)IP(G) and IP(G′) = (−1)e(G)VCP(G) where
e(G) is the number of edges of G. By applying this transformation one more time to G′, we obtain a graph G′′such that:
IP(G′′) = (−1)e(G′)VCP(G′) = (−1)e(G)IP(G)
Thus, the transformation of G into G′′ consists of the replacement of each
edge (u, v) in G by the subgraph represented in Figure 1.
v u
Fig. 1. The transformation of Lemma 3
In Theorem 5 we introduced a family (G′
n) such that G′nhas an even number
of edges, and IP(G′
n) = PER∗n. By applying the transformation above to (G′n),
we obtain a bipartite family (G′′
n) such that IP(G′′n) = PER∗n.
In the following we will not only use the statement of this lemma: we will also use the structure of G′′
sets of G′′
n. We will use for instance the fact that in one of those two sets, say
V1, all vertices have weight −1.
It is pointed out in [11] that, given a bipartite graph, one can construct naturally a partially ordered set. From the bipartite graph G′′
n = (V1, V2, E), we
define the partially ordered set (Xn, ≤) with Xn= V1∪ V2, and given x and y
in Xn, x ≤ y if and only if x ∈ V1, y ∈ V2 and (x, y) ∈ E. We see easily that ≤
is transitive and antisymmetric.
Next we recall the definition of an antichain.
Definition 9 (Antichain). An antichain A in a poset (X, ≤) is a subset of X
such that for all pair (x, y) of distinct elements of A, x and y are incomparable.
We define the antichain polynomial of a (weighted) poset (X, ≤) as the polyno-mial: AP(X) =X A Y x∈A w(x)
where the sum runs over all antichains A of (X, ≤). Let us consider a bipartite graph G and its corresponding poset (X, ≤). A set S ⊆ X is an antichain in (X, ≤) if and only if it is independent in G. We thus have: AP(X) = IP(G). Thus, we can identify the families (AP(Xn)) and (IP(G′′n)). We then define the
notion of ideal in a poset.
Definition 10 (Ideal). An ideal I in a poset (X, ≤) is a subset of X such that
for all x ∈ I, all y such that y ≤ x belong to I.
We can also define the ideal polynomial IPP(X) in a poset (X, ≤): IPP(X) =X
I
Y
x∈I
w(x)
where the sum runs over all ideals I of (X, ≤).
Given an ideal I in a poset (X, ≤), the maximal elements of I form an antichain A: since they are maximal in I, they cannot be compared. Conversely, given an antichain A, the set of elements x that are not greater than an element of A form an ideal. One can verify easily that those transformations are bijective and inverse of each other. We thus have a bijection between the ideals and the antichains of a given poset. This fact suffices to the authors of [11], since the bijection shows that a poset has the same number of antichains and ideals; the counting problems are thus equivalent.
But for our weighted counting problems, since the ideals and the antichains have different weights, we cannot identify simply AP(X) and IPP(X) for any poset X.
We do not know how to reduce a family of antichain polynomials into ideal polynomials in general, but in the case of the family (AP(Xn)), since the
struc-ture of the family (G′′
n) is particular, the problem is easier. We claim the
Theorem 6. For all integers n, we have: AP(Xn) = IPP(Xn)
The proof will be given at the end of this section.
Corollary 1. There exists a VNP-complete family of polynomials of the form (IPP(Xn)).
To conclude, we note that the ideal polynomial in a poset (X, ≤) may be ex-pressed as a polynomial associated to a S-formula. Namely, we associate to each xi ∈ X a boolean variable εi with the intended meaning that xi belongs to an
ideal when εiis true. For every pair (xi, xj) ∈ X such that xi≤ xj, the condition
xj∈ I ⇒ xi∈ I may be expressed by (εj⇒ εi), or (εi∨ εj). Thus, we have
IPP(X) = X ε∈{0,1}|X| ^ (i,j):xi≤xj εi∨ εjX ε , and as a result:
Theorem 7. There exists a VNP-complete family of polynomials associated to
a p-family of {OR1}-formulas.
To complete this section, we now provide the proof of Theorem 6. Let us fix an integer n. We recall that in the bipartite graph G′′
n= (V1, V2, E) constructed in
Lemma 3, each vertex of V1has weight −1. We also know that |V1| is even, since
the elements of V1are added two by two in the transformation from G′n to G′′n.
Fortunately, by modifying the correspondence between antichains and ideals, we can preserve the weights: we will construct in Lemma 4 a bijection from the antichains to the ideals of Xn that preserves the weights, and thus we have:
AP(Xn) = IPP(Xn).
Lemma 4. There exists a bijection (different from the natural one considered
previously) from the antichains to the ideals of Xn, this one keeping the weights
unchanged.
Proof. To an antichain A of Xn, we associate the set I such that:
– A and I coincide on V2.
– I ∩ V1 is the complement of A ∩ V1 in V1.
The map A 7→ I is clearly injective, and one can verify that the image I is an ideal: given x ∈ Xn and y ∈ I such that x ≤ y, we have that x ∈ V1 and
y ∈ V2. Therefore, y ∈ A, and x cannot belong to A as the elements of A
are incomparable. Thus, x belong to I. Our map is thus a bijection from the antichains to the ideals of Xn.
Since all the elements of V1 have weight −1 and |V1| is even, the weights of
5
Non affine constraints
In this section we consider the general case of a set S containing non affine constraints:
Theorem 8. For every set S containing a non affine relation, there exists a VNP-complete family (under p-projection) of polynomials associated to S-formulas.
The proof of this result is an analogue of the proof of the #P-completeness of the corresponding counting problems given in [5]. We will adapt this proof to our context. The authors use the notion of perfect and faithful implementation (definition 5.1 in [5]):
Definition 11. A conjunction of α boolean constraints {f1, . . . , fα} over a set
of variables x = {x1, . . . , xn} and y = {y1, . . . , yn} is a perfect and faithful
implementation of a boolean formula f (x), if and only if
1. for any assignment of values to x such that f (x) is true, there exists a unique assignment of values to y such that all the constraints fi(x, y) are satisfied.
2. for any assignment of values to x such that f (x) is false, no assignment of values to y can satisfy more than (α − 1) constraints.
We refer to the set x as the function variables and to the set y as the auxiliary variables.
We say, that a set S of logical relations implements perfectly and faithfully a boolean formula f (x) if there is a S-formula that implements f (x) perfectly and faithfully. We also extend the definition to logical relations: a set S of logical relations implements perfectly and faithfully a logical relation f if S implements perfectly and faithfully every application of f to a set of variables x.
Let us denote F the unary relation F (x) = x. From [5], lemma 5.30, we have: Lemma 5. If a logical relation f is not affine, then {f, F} implements at least
one of the three logical relations OR0, OR1 or OR2 perfectly and faithfully.
The following lemma, analogue to lemma 5.15 from [5], shows that perfect and faithful implementation provide a mechanism to do projections from the polynomials associated to sets of logical relations.
Lemma 6. Let S and S′ be two sets of logical relations such that every relation
of S can be perfectly and faithfully implemented by S′. Then every p-family of
polynomials associated to a p-family of S-formulas is a projection of a p-family of polynomials associated to a p-family of S′-formulas.
Proof. Let (φn) be a p-family of S-formulas, and let us fix an integer n.
Let x = {x1, . . . , xp} be the set of variables of the formula φn. This formula φn
is a conjunction of logical relations fi∈ S applied on variables from {x1, . . . , xp}.
If we replace each of those relations fi by a perfect and faithful implementation
using constraints in S′, using for each f
obtain a conjunction ψnof logical relations from S′applied on variable set x ∪ y,
where y = {y1, . . . , yq} is the union of the auxiliary variables sets added for each
logical relation fi.
Since all implementations are perfect and faithful, every assignment to x that satisfies all constraints of φn can be extended by a unique assignment to x ∪ y
that satisfies all constraints of ψn. Conversely, for an assignment to x that does
not satisfy all constraints of φn, no assignment to x ∪ y can extend the previous
one and satisfy every constraint of ψn.
Since ψn is a conjunction of logical relations from S′ applied on a set of
variables x ∪ y, ψnis a S′-formula. Furthermore, the number of constraints of ψn
is bounded by the product of the number of constraints of φn and the maximum
number of logical relations from S′ needed to implement a logical relation from
S - which does not depend on n. The size of ψn is therefore polynomially linear
in the size of φn. We have:
P (φn)(X1, . . . , Xp) = X ε∈{0,1}p φn(ε)X ε = X ε∈{0,1}p,y∈{0,1}q ψn(ε, y)X ε = X ε∈{0,1}p,y∈{0,1}q ψn(ε, y)X ε 1y1. . . 1yq = P (ψn)(X1, . . . , Xp, 1, . . . , 1)
Finally, the family (P (φn)) is a projection of the family (P (ψn)), which is a
p-family of polynomials associated to S′-formulas.
From the two previous lemmas, and from the VNP-completeness of families of polynomials associated to {OR0}- , {OR1}- and {OR2}-formulas, we conclude
that for every set of logical relations S such that S contains non affine rela-tions, there exists a VNP-complete family of polynomials associated to S ∪ {F}-formulas. To get rid of the logical relation {F}, the authors of [5] need to re-investigate the expressiveness of a non affine relation, and distinguish various cases. For our polynomial problems, we can easily force a boolean variable to be set to false by giving to the associated polynomial variable the value 0. We can now give the proof of Theorem 8:
Proof. Let (φn) be a p-family of S ∪ {F}-formulas such that (P (φn)) is
VNP-complete. The existence of such a family is ensured by lemmas 5 and 6.
Let us consider an integer n. φn(x1, . . . , xn) is a conjunction of logical
rela-tions from S applied to variables from x and and constraints of the form (xi = 0).
We remark, that if φn(x) contains the constraint (xi = 0), then the variable Xi
does not appear in the polynomial P (φn)(X1, . . . , Xn): all the monomials
con-taining the variable Xihave null coefficients. If we suppress from the conjunction
we obtain exactly the same polynomial: the monomials such that Xiappears in
it have null coefficients; the others correspond to assignments such that xi= 0.
Let us denote ψn the formula obtained by suppressing from φn all the
con-straints of the form (xi = 0). Since P (φn)(X1, . . . , Xn) = P (ψn)(y1, . . . , yn),
where yi is 0 if the constraint (xi = 0) was inserted in φn, and Xi otherwise,
(P (φn)) is a p-projection of (P (ψn)). Thus, the family (P (ψn)) is VNP-complete.
6
Affine relations with at least three variables
Here we consider the case of a set S containing large affine constraints. We first establish the existence of a VNP-complete family of polynomials associated to a p-family of affine formulas, and then show how to reduce this family to each affine constraint with at least three variables. In this section, our VNP-completeness results are in the sense of c-reduction.
Let us consider the n × n permanent PERn(M ) of a matrix M = (Mi,j).
It may be expressed as the polynomial associated to the formula accepting the n × n permutation matrices: PERn(M ) =Pεφn(ε)X
ε
This formula φn expresses, that each row and each column of the matrix ε
contains exactly one 1. Let us consider the formula ϕn defined by:
ϕn(ε) = n ^ i=1 εi1⊕ . . . ⊕ εin= 1 ∧ n ^ j=1 ε1j⊕ . . . ⊕ εnj = 1
The formula ϕn expresses, that each row and each column of ε contains an
odd number of values 1. Thus, ϕn accepts the permutation matrices, and other
assignments that contain more values 1. We therefore remark, that the n × n permanent is exactly the homogeneous component of degree n of P (ϕn). But
from Lemma 1, this implies a c-reduction from the permanent family to the p-family (P (ϕn)). Thus:
Lemma 7. The family (P (ϕn)) is VNP-complete with respect to c-reductions.
Through c-reductions and p-projections, this suffices to establish the exis-tence of VNP-complete families for affine formulas of at least three variables: Theorem 9. 1. There exists a VNP-complete family of polynomials associated
to {x ⊕ y ⊕ z = 0}-formulas.
2. There exists a VNP-complete family of polynomials associated to {x⊕y ⊕z =
1}-formulas.
3. For every set S containing an affine formula with at least three variables, there exists a VNP-complete family of polynomials associated to S-formulas. Proof. 1. Let us consider the formula ϕn. This formula is a conjunction of
affine relations with constant term 1: x1+ . . . + xk = 1. Let ϕ′n be the
formula obtained from ϕn by adding a variable a and replacing such clauses
degree 1 in the variable associated to a is exactly the polynomial P (ϕn):
when a is assigned to 1, the satisfying assignments of ϕ′
n are equal to the
satisfying assignments of ϕn. Since this term of degree 1 can be recovered by
polynomial interpolation of P (ϕ′
n), the family (P (ϕn)) c-reduces to (P (ϕ′n)).
ϕ′
n is a conjunction of affine relations with constant term 0. The polynomial
P (ϕ′
n) is the projection of the polynomial P (ψn), where the formula ψn is
obtained from ϕ′
nby replacing each affine relation of the type x1⊕. . .⊕xk = 0
by the conjunction of relations
(x1⊕ x2⊕ a1= 0) ∧ (a1⊕ x3⊕ a2= 0) ∧ . . . ∧ (ak−2⊕ xk−1⊕ xk = 0)
where the aiare new variables. In fact, on sees easily, that for a given
assign-ment of the xi satisfying ϕ′n, a single assignment of the aigives a satisfying
assignment of ψn; and that if the xi do not satisfy ϕ′n, no assignment of
the ai works on. The polynomial P (ϕ′n) is thus the polynomial obtained by
replacing the variables associated to ai by the value 1 in P (ψn); the family
(P (ϕ′
n)) is a p-projection of (P (ψn)).
2. The formula ψn constructed above is a conjunction of relations of the type
x ⊕ y ⊕ z = 0. Let us construct a new formula ψ′
n by introducing two new
variables a and b and replacing each of such relations by the conjunction (x ⊕ y ⊕ a = 1) ∧ (a ⊕ z ⊕ b = 1). One sees easily, that P (ψn) is the projection
of P (ψ′
n) obtained by setting the variables associated to a and b to 1 and 0
respectively.
3. Let us suppose, that S contains a relation of the type x1⊕ . . . ⊕ xk= 0, with
k ≥ 3. The polynomial P (ψn) is the projection of the polynomial associated
to the S-formula obtained by replacing each relation x ⊕ y ⊕ z = 0 of ψn
by a relation x ⊕ y ⊕ z ⊕ a1⊕ . . . ⊕ ak−3 = 0, and setting the variables
associated to the ai to 0. Thus, the family (P (ψn)) projects on a family
of polynomials associated to S-formulas, which is therefore VNP-complete. When S contains a relation with constant term 1, one projects the family (P (ψ′
n)) similarly.
7
#P-completeness proofs
Up to now, we have studied vertex weighted graphs mostly from the point of view of algebraic complexity theory. Putting weights on edges, or on vertices, can also be useful as an intermediate step in #P-completeness proofs [14,7]. Here we follow this method to obtain new #P-completeness results. Namely, we prove #P-completeness under many-one reductions for several problems which were only known to be #P-complete under oracle reductions.
Theorem 10. The following problems are #P-complete for many-one
reduc-tions.
1. Vertex Cover: counting the number of vertex covers of a given a graph. 2. Independent Set: counting the number of independent sets of a given graph.
3. Bipartite Vertex Cover: the restriction of vertex cover to bipartite graphs. 4. Bipartite Independent Set: the restriction of independent set to bipartite
graphs.
5. Antichain: counting the number of antichains of a given poset. 6. Ideal: counting the number of ideals of a given poset.
7. Implicative 2-SAT: counting the number of satisfying assignments of a con-junction of implicative 2-clauses.
8. Positive 2-SAT: counting the number of satisfying assignments of a conjunc-tion of positive 2-clauses.
9. Negative 2-SAT: counting the number of satisfying assignments of a conjunc-tion of negative 2-clauses.
Remark 2. #P-completeness under oracle reductions is established in [11] for
the first six problems, in [9] for the 7th problem and in [15] for the last two. In Section 2, the last three problems are denoted #SAT(S) where S is respectively equal to {OR1}, {OR0} and {OR2}.
Proof. Provan and Ball establish in [11] the equivalence of Problems 1 and 2, 3
and 4, and 5 and 6; they produce many-one reductions from 1 to 8 and from 4 to 5, and Linial gives in [9] a many-one reduction from 6 to 7. Problems 8 and 9 are clearly equivalent. Therefore, to obtain #P-completeness under many-one reductions for all those problems, we just need to show the #P-completeness of Problem 1 and to produce a many-one reduction from Problem 1 to Problem 3 (replacing the oracle reduction from [11]).
In order to prove the #P-completeness of Problem 1, we first establish a many-one reduction from the #P-complete problem of computing the perma-nent of {0, 1}-matrices (which is known to be #P-complete under many-one reductions [16]) to the problem of computing the vertex cover polynomial of a weighted graph with weights in {0, 1, −1}. In [3], B¨urgisser attributes to Jer-rum a projection from the permanent to the partial permanent, with the use of the constant −1. Applied to a {0, 1}-matrix, this gives a many-one reduction from the permanent on {0, 1}-matrices to the partial permanent on {0, 1, −1}-matrices. By Theorem 5, the n×n partial permanent is equal to the independent set polynomial of the graph G′
n; the reduction is obviously polynomial.
More-over, by Lemma 2 this polynomial is the projection of the vertex cover polyno-mial of Gn, with the use of the constant −1. The partial permanent on entries
in {0, 1, −1} therefore reduces to the vertex cover polynomial on graphs with weights in {0, 1, −1}.
Let G be such a vertex weighted graph, with weights in {0, 1, −1}. A vertex cover of nonzero weight does not contain any vertex v of weight 0, and in order to cover the edges that are incident to v, it must contain all its neighbors. One can therefore remove v, and replace each edge from v to another vertex u by a self-loop (an edge from u to u). Thus, we obtain a graph G′ with weights in
{1, −1} such that VCP(G) = VCP(G′).
To deal with the weights −1, we use a method similar to [14]. Since VCP(G′)
construct an integer N and a graph H such that the number of vertex covers of H modulo N is equal to VCP(G′). This will establish a reduction from the
boolean permanent to counting vertex covers.
We choose N larger than the maximum value of the number of vertex covers of G′: N = 2v(G′)
+ 1 will suit our purposes. Now that we compute the number of vertex covers modulo N , we can replace each −1 weight in G′ by the weight
N − 1 = 2v(G′). But one can simulate such a weight on a vertex by adding to it
v(G′) leaves. ⇔ u w(u) = 2k u l1 l2 · · · lk
Fig. 2. Simulation of a weight 2k
Finally, we construct a many-one reduction from vertex cover to bipartite vertex cover. From Lemma 3, we have a projection from the vertex cover poly-nomial of a graph to the vertex cover of a bipartite graph, with the use of −1 weights. To eliminate these weights, we can follow the method used in our above proof of the #P-completeness of Problem 1. Indeed, since the leaves added to the graph preserve bipartiteness, we obtain a reduction from counting vertex covers in a general graph to counting vertex covers in a bipartite graph. The proof of Creignou and Hermann’s dichotomy theorem [4,5] is based on many-one reductions from the last 3 problems of Theorem 10. We have just shown that these 3 problems are #P-complete under many-one reductions. As a result, we have the following corollary to Theorem 10.
Corollary 2. Theorem 1 still holds for #P-completeness under many-one
re-duction.
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